Abstract

This thesis consists of an introduction and four independent chapters.

In Chapter 1, we study homeomorphism groups of metrizable compactifications of the natural numbers. Those groups can be represented as almost zero-dimensional Polishable subgroups of the group S∞. We show that all Polish groups are continuous homomorphic images of almost zero-dimensional Polishable subgroups of S∞. We also find a sufficient condition for these groups to be one dimensional.

In Chapter 2, we study the connections between properties of the action of a countable group Γ on a countable set X and the ergodic theoretic properties of the corresponding shift action of Γ on MX, where M is a measure space. In particular, we show that the action of Γ on X is amenable iff the corresponding shift has almost invariant sets. This is joint work with Alexander Kechris.

In Chapter 3, we prove that if the Koopman representation associated to a measure-preserving action of a countable group on a standard non-atomic probability space is non-amenable, then there does not exist a countable-to-one Borel homomorphism from its orbit equivalence relation to the orbit equivalence relation of any modular action (i.e., an action on the boundary of a countably splitting tree), generalizing previous results of Hjorth and Kechris. As an application, for certain groups, we connect antimodularity to mixing conditions. This is joint work with Inessa Epstein.

In Chapter 4, we study full groups of countable, measure-preserving equivalence relations. Our main results include that they are all homeomorphic to the separable Hilbert space and that every homomorphism from an ergodic full group to a separable group is continuous. We also find bounds for the minimal number of generators of a dense subgroup of full groups allowing us to distinguish full groups of equivalence relations generated by free, ergodic actions of the free groups Fn and Fm if m and n are sufficiently far apart. We also show that an ergodic equivalence relation is generated by an action of a finitely generated group iff its full group has a finitely generated dense subgroup. This is joint work with John Kittrell.